A Cartesian-Grid Discretisation Scheme Based on Local ......Lo cal grid Control olumev Boundary Boundary Figure 1: Local networks in x 1 (top) and x 2 (bottom) (: RBF centre and o:
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Copyright © 2009 Tech Science Press CMES, vol.51, no.3, pp.213-238,
2009
A Cartesian-Grid Discretisation Scheme Based on Local Integrated
RBFNs for Two-Dimensional Elliptic Problems
N. Mai-Duy1 and T. Tran-Cong1
Abstract: This paper reports a new numerical scheme based on
Cartesian grids and local integrated radial-basis-function networks
(IRBFNs) for the solution of second-order elliptic differential
problems defined on two-dimensional regular and irregular domains.
At each grid point, only neighbouring nodes are activated to
construct the IRBFN approximations. Local IRBFNs are introduced
into two dif- ferent schemes for discretisation of partial
differential equations, namely point col- location and
control-volume (CV)/subregion-collocation. Linear (e.g. heat flow)
and nonlinear (e.g. lid-driven triangular-cavity fluid flow)
problems are consid- ered. Numerical results indicate that the
local IRBFN CV scheme outperforms the local IRBFN point-collocation
scheme regarding accuracy. Moreover, the former shows a similar
level of the matrix condition number and a significant improvement
in accuracy over a linear CV method.
Keywords: local approximations, integrated RBFNs, point
collocation, subre- gion collocation, second-order differential
problems.
1 Introduction
RBF-based discretisation methods have emerged as a new attractive
solver for par- tial differential equations (PDEs) (e.g. [Fasshauer
(2007)]). They have the capabil- ity to work well for problems
defined on irregular domains. Very accurate results can be achieved
using only a relatively-small number of nodes. However, RBF
matrices are dense and generally ill-conditioned. To resolve this
problem, local RBF methods have been developed, resulting in having
to solve a sparse system of algebraic equations. The RBF
approximations are constructed locally on small overlapping regions
which are represented by a set of structured points or a set of
scattered points. Works reported include [Lee, Liu, and Fan (2003);
Shu, Ding, and Yeo (2003); Tolstykh and Shirobokov (2003);
Chantasiriwan (2004); Shu, Ding,
1 Computational Engineering and Science Research Centre, Faculty of
Engineering and Surveying, University of Southern Queensland,
Toowoomba, QLD 4350, Australia.
214 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
and Yeo (2005); Tolstykh and Shirobokov (2005); Wright and Fornberg
(2006); Kosec and Sarler (2008a); Kosec and Sarler (2008b); Orsini,
Power, and Morvan (2008); Sanyasiraju and Chandhini (2008); Vertnik
and Sarler (2009)].
To transform a PDE into a set of algebraic equations, one needs to
discretise the problem domain. For irregular domains, this task can
be expensive and time- consuming. It can be seen that using
Cartesian grids to represent the domain is economical. Considerable
effort has been put into the development of Cartesian- grid-based
computational techniques (e.g. [Johansen and Colella (1998); Jomaa
and Macaskill (2005); Hu, Young, and Fan (2008); Parussini and
Pediroda (2008); Pasquim and Mariani (2008); Bourantas, Skouras,
and Nikiforidis (2009)]).
The proposed numerical procedure combines strengths of the local
RBF approach and the Cartesian-grid approach for solving 2D
differential problems. At each grid point, only neighbouring nodes
are activated to construct the RBF approximations. Unlike local RBF
techniques reported in the literature, RBFNs are employed here to
approximate highest-order derivatives in a given PDE and
subsequently integrated to obtain expressions for lower-order
derivatives and the field variable [Mai-Duy and Tran-Cong (2001)].
This use of integration to construct the approximations provides an
effective way to circumvent the problem of reduced convergence rate
caused by differentiation and to implement derivative boundary
conditions (e.g. [Mai-Duy (2005); Mai-Duy and Tran-Cong (2006)]).
In this study, we introduce lo- cal integrated RBFNs into two PDE
discretisation formulations, namely point col- location and
control-volume (CV)/subregion-collocation, and then conduct some
numerical experiments to investigate accuracy of the two local
IRBFN techniques.
Recirculating viscous flows in enclosed cavities have received a
great deal of at- tention in fluid mechanics community as they can
produce interesting flow features at different Reynolds numbers.
Examples of this type include lid-driven flows in square and
triangular cavities. For such problems, at the two top corners, the
ve- locity is discontinuous and the stress is unbounded. These pose
great challenges for numerical simulation. In contrast to the
square cavity problem, the triangular cavity flow presents a severe
test for structured-grid-based numerical methods (e.g. [Jyotsna and
Vanka (1995)]). The latter is chosen here to investigate the
perfor- mance of the present local IRBFN CV technique. The flow is
simulated using the streamfunction and vorticity formulation
discretised on rectangular grids. Attrac- tive features of the
proposed technique are (i) no coordinate transformations are
required and (ii) computational boundary conditions for the
vorticity are derived in a new effective way, where analytic
formulae that need only nodal values of the streamfuntion on one
grid line [Le-Cao, Mai-Duy, and Tran-Cong (2009)] are
utilised.
The remainder of this paper is organised as follows. A brief review
of integrated
Cartesian-Grid Local-IRBFN Scheme 215
RBFNs is given in Section 2. The proposed computational procedure
is presented in Section 3 and numerically verified through a series
of examples in Section 4. Section 5 concludes the paper.
2 Integrated radial-basis-function networks
RBFNs allow a conversion of a function f from a low-dimensional
space (e.g. 1D- 3D) to a high-dimensional space in which the
function can be expressed as a linear combination of RBFs
f (x) = m
∑ i=1
w(i)G(i)(x), (1)
where the superscript (i) is the summation index, x the input
vector, m the number of RBFs, {w(i)}m
i=1 the set of network weights to be found, and {G(i)(x)}m i=1 the
set
of RBFs.
This study is concerned with second-order differential problems in
two dimen- sions. The integral approach [Mai-Duy and Tran-Cong
(2001)] uses RBFNs (1) to represent the second-order derivatives of
the field variable u in a given PDE. Ap- proximate expressions for
the first-order derivatives and the variable itself are then
obtained through integration as
∂ 2u(x) ∂x2
u[x j](x) = m
∑ i=1
w(i) [x j]
H(i) [x j](x)+ x jC1[x j](xk)+C2[x j](xk), (4)
where the subscript [x j] is used to denote the quantities
associated with the process of integration with respect to the x j
variable; C1[x j](xk) and C2[x j](xk) the constants of integration
which are univariate functions of the variable other than x j (i.e.
xk
with k 6= j); H(i) [x j]
(x) = ∫
∫ H(i)
[x j] (x)dx j. The reader is
referred to [Mai-Duy and Tran-Cong (2001); Mai-Duy and Tanner
(2005); Mai- Duy (2005);Mai-Duy and Tran-Cong (2005)] for further
details (e.g. explicit forms of integrated and differentiated
RBFs).
3 Proposed technique
The 2D problem domain is discretised using a Cartesian grid.
Boundary points are generated through the intersection of the grid
lines and the boundaries. For a ref-
216 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
erence point, we form two local integrated networks using (2)-(4):
one associated with the x1 coordinate and the other with the x2
coordinate. The two networks are constructed on the same set of l×
l grid lines. The reference point may not be the centre of the
local grid when the construction process is carried out near the
boundary (Figure 1).
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
PSfrag repla ements Lo al gridControl volumeBoundary −0.3 −0.25
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
PSfrag repla ementsLo al gridControl volumeBoundary Lo al
gridControl volumeBoundary Figure 1: Local networks in x1 (top) and
x2 (bottom) (∗: RBF centre and o: interior point).
For local grids entirely embedded in the domain, the two networks
have the same set of RBF centres which are chosen to be the
interior grid nodes. The value of m in (4) is equal to l2.
For local grids that are cut by irregular boundary, one generally
has different sets of RBF centres for the two associated networks.
A set of the RBF centres for the x j
network is comprised of the interior grid nodes and the boundary
nodes generated by the x j grid lines. The value of m in (4) may be
less than l2 (Figure 1).
We also employ IRBFNs to represent the variation of the constants
of integration. The construction process for C1[x j](xk) is exactly
the same as that for C2[x j](xk). To simplify the notation, the
subscripts 1[x j] and 2[x j] are dropped. The function C(xk)
Cartesian-Grid Local-IRBFN Scheme 217
w(i)h(i)(xk)+ xkc1 + c2, (7)
where c1 and c2 are the constants of integration which are simply
unknown values, and g(i), h(i) and h(i) the one-dimensional forms
of G(i), H(i) and H(i), respectively. Collocating (7) at the local
grid points x(i)
k with i = {1,2, · · · , l} leads to
C = T w, (8)
where C and w are the vectors of length l and (l + 2),
respectively, and T is the transformation matrix of dimensions l×
(l +2)
C = (
.
Taking (8) into account, the value of C in (7) at an arbitrary
point xk can be com- puted in terms of nodal values of C as
C(xk) = [ h(1)(xk), h(2)(xk), · · · , h(l)(xk),xk,1
C(xk) = l
∑ i=1
P(i)(xk)C(i), (10)
where P(i)(xk) is the product of the first vector on RHS and the
ith column of T +, and T + is the generalised inverse of T of
dimensions (l + 2)× l, which can be
218 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
obtained using the SVD technique. It is noted that both P(i)(xk)
and T + could not be defined explicitly.
Substitution of (10) into (3) and (4) yields
∂u(x) ∂x j
, (11)
∑ i=1
+ l
. (12)
For convenience of presentation, expressions (2), (11) and (12) can
be rewritten as
∂ 2u(x) ∂x2
∑ i=1
where
2l i=1,
{H(i) [x j]
(i) [x j]
i=m+l+1 ≡ {0} l i=1,
{H(i) [x j](x)}m+l
i=m+1 ≡ {x jP (i) [x j]
(xk)}l i=1, {H(i)
[x j](x)}m+2l i=m+l+1 ≡ {P
(i) [x j]
i=m+1 ≡ {C (i) 1[x j] }l
i=1, and {w(i) [x j] }m+2l
i=m+l+1 ≡ {C (i) 2[x j] }l
i=1.
We seek the solution in terms of nodal values of the field variable
u. To do so, (15) is collocated at the nodal points on the local
grid, from which the relationship between the network-weight space
and the physical space can be established as
u[x j] = T[x j]w[x j], (16)
w[x j] = T + [x j]
u[x j], (17)
where u[x j] is the vector of length m consisting of the nodal
values of u on the local grid, w[x j] the vector of length (m + 2l)
made up of the RBF weights and
the nodal values of C(i) 1[x j]
and C(i) 2[x j]
, and T + [x j]
the generalised inverse of T[x j]. The
transformation matrix T[x j] has the entries T[x j]rs = H(s) [x
j](x
(r)), where 1 ≤ r ≤ m and 1≤ s≤ (m+2l).
Cartesian-Grid Local-IRBFN Scheme 219
It is noted that the two vectors, u[x1] and u[x2], are unknown.
From now on, they are forced to be identical
u[x1] ≡ u[x2] ≡ u. (18)
The values of u, ∂u/∂x j and ∂ 2u/∂x2 j at an arbitrary point x can
be computed in
terms of nodal variable values as
u(x) = 1 2
2
[x j] (x) ] T +
[x j] (x) ] T +
[x j] u, (20)
[x j] u. (21)
In (19), there are two integrated networks in the x1 and x2
directions that produce two values of the function at a point.
Theoretically, they are the same. However, due to approximation
errors, these two values are not identical. As a result, the
function value is computed by taking the average of the two.
For the point-collocation formulation, there are no integrations
required for the discretisation. The process of converting the PDE
into a set of algebraic equations is straightforward.
For the control-volume formulation, one has to define a control
volume for each node, over which the PDE will be integrated. The
control volume is formed using the lines that are parallel to the
x1 and x2 axes and go through the middle points between the
reference node and its neighbours or appropriate points on the
bound- ary (Figure 1). Integrals can be calculated using high-order
(e.g. 5-points used here) Gauss quadrature since the present
approximation scheme allows the accu- rate evaluation of the
variable u and its derivatives at any point within the local
grid.
The use of local integrated networks results in a sparse system of
simultaneous equations. It can be seen that operations on zero
elements are unnecessary. Avoid- ing these operations provides a
considerable saving in time. By taking account of sparseness of the
system matrix, one has the capability to reduce the computational
time and storage facilities. Such sparse equation sets can be
solved effectively by means of iterative solvers (e.g. generalised
minimum residual methods).
4 Numerical examples
For all numerical examples presented here, the approximations are
constructed on local grids of 5× 5. IRBFNs are implemented with the
multiquadric (MQ) basis
220 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
function whose form is given
G(i)(x) = √
(x− c(i))T (x− c(i))+a(i)2, (22)
where c(i) and a(i) are the centre and width of the ith MQ basis
function, respec- tively. The set of centres and the set of
collocation points are identical. It is well known that the
width/shape-parameter strongly affects accuracy of the RBF scheme.
However, it is still very difficult to determine the optimal value
of the shape parameter in practice. In this study, we do not focus
on the study of the RBF width. All MQ centres are associated with
the same width that is chosen to be the grid size. For problems
whose exact solution is available, we use the discrete relative L2
norm of u, denoted by Ne(u), to measure accuracy of an approximate
scheme. We apply the matrix 1-norm estimation algorithm for
estimating condi- tion numbers of the system matrix. Furthermore,
linear CV (central difference) techniques, which are similar to
those described in [Patankar (1980)], are referred to as a standard
CV technique.
4.1 Test problem
2 = 0 (23)
with Dirichlet boundary conditions. Two computational domains,
namely a unit square 0 ≤ x1,x2 ≤ 1 and a circle centered at the
origin with radius of 0.5, are considered. The exact solution is
given by
ue = 1
from which the boundary values of u can be derived.
The point-collocation formulation consists in forcing (23) to be
satisfied exactly at discrete points in order to form a determined
set of algebraic equations. It means that (23) needs be collocated
at the interior grid nodes.
For the control-volume formulation, (23) is forced to be satisfied
in the mean. In- tegrating (23) over a control volume i, we
have∫
i
(∇u ·n)dΓi = 0, (26)
Cartesian-Grid Local-IRBFN Scheme 221
where Γi is the boundary of i and n the outward normal unit vector.
To compute ∂u/∂x j on the faces that are parallel to the xk (k 6=
j) direction, we use the x j
network.
Uniform Cartesian grids are employed to represent the problem
domain.
Table 1: Rectangular domain: Condition numbers of the system matrix
by stan- dard CV, local IRBFN collocation and local IRBFN CV
methods. Notice that a(b) means a×10b.
Grid Standard-CV IRBFN-collocation IRBFN-CV 15×15 1.1(2) 1.8(2)
1.2(2) 27×27 3.9(2) 6.4(2) 4.3(2) 39×39 8.5(2) 1.3(3) 9.3(2) 51×51
1.4(3) 2.4(3) 1.6(3) 63×63 2.2(3) 3.7(3) 2.4(3) 75×75 3.2(3) 5.2(3)
3.5(3) 87×87 4.3(3) 7.1(3) 4.7(3) 99×99 5.6(3) 9.2(3) 6.2(3)
111×111 7.1(3) 1.1(4) 7.8(3) 123×123 8.7(3) 1.4(4) 9.6(3) 135×135
1.0(4) 1.7(4) 1.1(4) 147×147 1.2(4) 2.0(4) 1.3(4) 159×159 1.4(4)
2.4(4) 1.6(4) 171×171 1.7(4) 2.7(4) 1.8(4) 183×183 1.9(4) 3.1(4)
2.1(4) 195×195 2.2(4) 3.6(4) 2.4(4) 207×207 2.5(4) 4.0(4)
2.7(4)
In the case of rectangular domain, condition numbers of the system
matrix by the present local collocation and CV techniques are
presented in Table 1. Results ob- tained by the standard CV method
are also included for comparison purposes. It can be seen that the
three methods yield a similar level of the matrix condition num-
ber. The use of local approximations leads to a significant
improvement in stability over that of global approximations. It was
reported in the literature that the global RBF matrices may be
ill-conditioned when using 1000 nodes. Here, with 42849 nodes
taken, condition numbers of the RBF matrix are only O(104). In
terms of accuracy, both RBF methods are more accurate than the
standard CV method as
222 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
10 −3
10 −2
10 −1
10 0
10 −7
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
Standard CVM
IRBFN Collocation
IRBFN CVM
PSfrag repla ements hNe(u) Figure 2: Rectangular domain, [7× 7,11×
11, · · · ,203× 203]: Error versus grid size for standard CVM/FDM
and local IRBFN methods.
shown in Figure 2. The IRBFN-CV method outperforms the
IRBFN-collocation method. Given a grid size, the CPU time for the
IRBFN-CVM solution is seen to be greater than that for the
standard-CVM solution. However, from Figure 2, the IRBFN-CVM is
much more accurate than the standard CVM. To achieve a similar
level of accuracy, it is necessary to use denser grids for the
standard CVM. For example, to yield Ne = 1.9× 10−7, one needs to
employ approximately a grid of 1701× 1701 for the standard CVM
(this grid density is estimated through extrap- olation) and only
203× 203 for the IRBFN CVM. It is noted that very high grid
densities lead to ill-conditioned matrices. For a given accuracy,
the IRBFN CVM can thus be more efficient than the standard CVM.
Figure 3 shows the locations of nonzero entries in the IRBFN system
matrix.
In the case of circular domain, we generate boundary nodes through
the intersection of the grid lines and the boundary. It can be seen
that there may be some interior grid nodes that are very close to
the boundary. A parameter = h/8 is introduced here. Interior nodes,
which fall within a small distance to the boundary, are set aside.
The matrix condition number and the accuracy of the three methods
are shown in Table 2 and Figure 4. Remarks for this case are
similar to those for the rectangular case.
Theoretical studies [Sarra (2006)] showed that IRBFNs have higher
approximation
Cartesian-Grid Local-IRBFN Scheme 223
x 10 4
nz = 549081
Figure 3: Rectangular domain, 151× 151: The structure of the 22201×
22201 IRBFN system matrix.
power than differentiated RBFNs. The implementation of local- and
global-IRBFN versions incorporating Cartesian grids shares many
common features. The latter was presented in detail in our previous
works [Mai-Duy and Tran-Cong (2009a); Mai-Duy and Tran-Cong
(2009b)]. For the handling of a Neumann boundary con- dition on a
curved boundary, the reader is referred to [Mai-Duy and Tran-Cong
(2009b)] for a full detail. Generally speaking, global versions are
capable of giv- ing more accurate results than local versions.
However, global schemes produce fully-populated matrices that may
limit the number of nodes to a few hundreds only. For problems,
which require a relatively-dense discretisation for an accurate
simulation, the use of local schemes is a preferred option.
Numerical experiments studied here indicate that the control volume
formulation works better for local IRBFNs than the collocation
formulation. The IRBFN-CV method is now applied to simulate some
heat transfer and fluid flow problems.
224 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
Table 2: Circular domain: Condition numbers of the system matrix by
standard CV, local IRBFN collocation and local IRBFN CV methods.
Notice that a(b) means a×10b.
Grid Standard-CV IRBFN-collocation IRBFN-CV 15×15 9.8(1) 2.1(2)
1.0(2) 27×27 3.3(2) 1.0(3) 3.7(2) 39×39 8.5(2) 3.2(3) 8.6(2) 51×51
1.3(3) 4.7(3) 1.4(3) 63×63 2.2(3) 8.1(3) 2.4(3) 75×75 2.8(3) 8.3(3)
3.2(3) 87×87 3.8(3) 1.1(4) 4.4(3) 99×99 5.6(3) 1.8(4) 7.0(3)
111×111 6.5(3) 1.9(4) 7.4(3) 123×123 8.8(3) 2.7(4) 1.1(4) 135×135
1.0(4) 3.5(4) 1.1(4) 147×147 1.3(4) 4.6(4) 1.6(4) 159×159 1.5(4)
5.8(4) 1.7(4) 171×171 1.8(4) 6.5(4) 2.4(4) 183×183 2.0(4) 7.5(4)
2.2(4) 195×195 2.2(4) 6.6(4) 2.3(4) 207×207 2.7(4) 8.5(4)
3.2(4)
4.2 Heat flow
∇.
) = 0, x ∈, (27)
where v is a prescribed velocity, the domain and Pe the Peclet
number. Here,
and v are taken as [0,1]× [−0.5,0.5] and (1,0)T , respectively.
Boundary conditions are prescribed as follows
θ = 0, for x2 =−0.5 and x2 = 0.5, (28)
θ = cos(πx2) for x1 = 0, and (29)
θ = 0 for x1 = 1. (30)
Cartesian-Grid Local-IRBFN Scheme 225
10 −3
10 −2
10 −1
10 0
10 −7
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
Standard CVM
IRBFN Collocation
IRBFN CVM
PSfrag repla ements hNe(u) Figure 4: Circular domain, [7× 7,11× 11,
· · · ,203× 203]: Error versus grid size for standard CVM/FDM and
local IRBFN methods.
The exact solution to this problem can be verified to be
θe = cos(πx2)
where a = 0.5 (
. This problem is taken from [Kohno and Bathe (2006)].
The temperature boundary layer becomes thinner with increasing Pe.
At Pe = 1000, very steep boundary layer is formed. Figure 5 shows
the temperature contours for three different values of Pe by the
present CV method. Its accuracy is better than that of the standard
CV method as shown in Table 3. Figure 6 displays variations of
temperature along the centre line. It can be seen that the proposed
method produces very accurate results for all cases. Figure 7 show
that there are no fluctuations in the IRBFN CVM solution.
4.3 Lid-driven triangular-cavity flow
Consider the steady recirculating flow of a Newtonian fluid in an
equilateral trian- gular cavity. Figure 8 shows the cavity
geometry. We take P =
√ 3 and Q = 3.
The lid moves from left to right with a unit velocity (v = (1,0)T
), while the left
226 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
PSfrag repla ements Pe = 10 (21 21)
PSfrag repla ements Pe = 100 (51 51)
PSfrag repla ements Pe = 1000 (401 401)
Figure 5: Heat flow: Temperature distribution for a wide range of
Pe by the local IRBFN-CV method. There are 21 contour lines whose
values vary linearly between the two extremes.
Cartesian-Grid Local-IRBFN Scheme 227
0.2
0.4
0.6
0.8
1
Exact solutionPSfrag repla ements x1 u
Figure 6: Heat flow: temperature variations on the horizontal
centreline for differ- ent values of Pe. Exact values are also
included for comparison purposes.
0.96 0.98 1 1.02 1.04 1.06
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
PSfrag repla ements x1 u
Figure 7: Heat flow: variations of temperature on the centreline in
the boundary layer by the two techniques for Pe = 1000 using the
same grid.
228 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
Table 3: Heat Flow, Pe = 1000: Error Ne(u) by standard CV and local
IRBFN CV methods. Notice that a(−b) means a×10−b.
Grid Standard-CV IRBFN-CV 11×11 2.69(-1) 1.00(-1) 51×51 1.83(-2)
3.69(-3)
101×101 4.25(-3) 9.36(-4) 151×151 1.83(-3) 3.47(-4) 201×201
1.01(-3) 1.75(-4) 251×251 6.45(-4) 1.11(-4) 301×301 4.46(-4)
8.32(-5) 351×351 3.27(-4) 6.92(-5) 401×401 2.50(-4) 6.15(-5)
PSfrag repla ements x1 x2
(0; 0) (P;Q)(P;Q)( = 0; =n = 0) ( = 0; =n = 0)
( = 0; =n = 1) Figure 8: Cavity flow: geometry and boundary
conditions.
and right walls are stationary (v = (0,0)T ). The problem was
numerically studied by different techniques, including the
finite-different method with a transformed geometry (e.g. [Ribbens,
Watson, and Wang (1994)] and the finite-element method with a
flow-condition-based interpolation (e.g. [Kohno and Bathe
(2006)]).
Cartesian-Grid Local-IRBFN Scheme 229
ω = ∇ 2 ψ, (32)
∇ 2 ω, (33)
where Re is the Reynolds number defined as Re = UL/ν (L: the
characteristic length, U : the characteristic velocity and ν the
kinematic viscosity), ψ the stream- function, ω the vorticity, v1 =
∂ψ/∂x2 and v2 = −∂ψ/∂x1. The reference length and velocity are
presently chosen as L = Q/3 and U = 1 (the velocity of the lid),
respectively.
Spatial discretisation: The equilateral triangular cavity is
discretised using a Cartesian grid. Grid nodes inside the cavity
are taken to be the interior grid points. For this particular
geometry, boundary nodes are generated using only grid lines that
contain at least one interior grid node. With this approach, the
set of RBF centres/collocation-points do not include the two top
corners and hence infinite values of the vorticity do not enter the
discrete system.
Boundary conditions: Boundary conditions for the velocity lead to ψ
= 0 and ∂ψ/∂x2 = 1 on the lid, and ψ = 0 and ∂ψ/∂n = 0 on the
remaining walls (n: the direction normal to the wall).
We will use the boundary values ψ = 0 for solving the
streamfunction equation. From ψ = 0 and ∂ψ/∂n = 0, one obtains
∂ψ/∂x1 = 0 and ∂ψ/∂x2 = 0 along the boundaries, which are used for
solving the vorticity equation. A vorticity boundary condition is
computed through
ωb = ∂ 2ψb
∂x2 1
, (34)
where the subscript b is used to indicate the boundary value. On
the top wall, (34) reduces to
ωb = ∂ 2ψb
∂x2 2
On the side walls, there are two possible cases
(i) A boundary point is also a grid node and the number of nodal
points on the two associated grid lines are sufficiently-large for
an accurate approximation, and
230 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
(ii) A boundary point is not a grid node or the number of nodal
points on one of the two associated grid lines is too small.
To compute ωb, one can use (34) directly for the first case but may
need to derive a suitable formula from (34), which requires
information about ψ on one grid line only, for the second case. The
latter will be handled here by applying formulae reported in our
previous work (e.g. [Le-Cao, Mai-Duy, and Tran-Cong (2009)]),
namely
ωb = [
1+ (
ωb = [
1+ (
, (37)
for a x2-grid line. In (36) and (37), t1 and t2 are the x1- and
x2-components of the unit vector tangential to the boundary.
As mentioned earlier, one has to incorporate ∂ψb/∂x1 = 0 and
∂ψb/∂x2 = 0 into (35), (34), (36) and (37). The incorporation
process is similar to that in [Ho-Minh, Mai-Duy, and Tran-Cong
(2009)]. It will be briefly reproduced here for the sake of
completeness. Consider a x j grid line. On the line, one has
∂ 2ψ(x) ∂x2
∂ψ(x j) ∂x j
ψ(x j) = n j
w(i)h(i)(x j)+ x jc1 + c2, (40)
where n j is the number of nodal points on the line and other
notations are de- fined as before. There are two extra coefficients
c1 and c2 in (40). As a re- sult, two extra equations representing
boundary derivative values ∂ψ(x(1)
j )/∂x j and
∂ψ(x(n j) j )/∂x j can be added to the transformation system
ψ
∂ψ(x(1) j )
∂x j
Cartesian-Grid Local-IRBFN Scheme 231
where ψ is the vector of nodal variable values of length n j, w the
coefficient vector of length (n j + 2) and T is the transformation
matrix of dimensions (n j + 2)× (n j +2)
ψ = (
, (42)
w = (
j , 1
j , 1 ...
j ), x(n j) j , 1
h(1)(x(1) j ), h(2)(x(1)
h(1)(x(n j) j ), h(2)(x(n j)
j ), · · · , h(n j)(x(n j) j ), 1, 0
.
The values of ∂ 2ψ/∂x2 j at the two boundary points can be computed
by ∂ 2ψ(x(1)
j ) ∂x2
j ), 0, 0
j ), · · · , g(n j)(x(n j) j ), 0, 0
] T −1
where T −1 is an inverse of T .
By means of point collocation and integration constants, derivative
boundary values are forced to be satisfied exactly. Moreover, all
grid points on the associated grid lines are used to compute ωb.
The present boundary schemes thus have a global property.
Four Cartesian grids, namely Grid 1 (2352 interior points), Grid 2
(5402 points), Grid 3 (9702 points) and Grid 4 (15252 points), are
employed to study the conver- gence of the solution. The flow is
simulated at the Reynolds number of 0, 100, 200 and 500. A
time-marching approach is applied here to solve the present system
of non-linear equations. For the vorticity transport equation (33),
the diffusive and convective terms are treated implicitly and
explicitly, respectively. We choose the
232 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
initial solution to be the solution at a lower Re. For the case of
Re = 0, the flow starts from rest.
PSfrag repla ements Re = 0, Grid 1
PSfrag repla ements Re = 100, Grid 2
PSfrag repla ements Re = 200, Grid 3
PSfrag repla ements Re = 500, Grid 4
Figure 9: Cavity flow: Streamlines which are drawn using 21 uniform
lines be- tween the minimum and zero values, and 11 uniform lines
between the zero and maximum values.
Figures 9 and 10 present contour plots of the streamfunction and
vorticity variables, which look reasonable when compared with those
available in the literature (e.g. [Ribbens, Watson, and Wang
(1994); Kohno and Bathe (2006)]).
Figure 11 presents variations of the x1 component of the velocity
vector on the vertical centreline x1 = 0 and the x2 component of
velocity on the horizontal line x2 = 2. Results obtained by [Kohno
and Bathe (2006)] are also included for com- parison purposes. It
can be seen that the present results agree well with those by the
flow-conditioned-based interpolation FEM for all values of
Re.
Cartesian-Grid Local-IRBFN Scheme 233
PSfrag repla ements Re = 0, Grid 1
PSfrag repla ements Re = 100, Grid 2
PSfrag repla ements Re = 200, Grid 3
PSfrag repla ements Re = 500, Grid 4
Figure 10: Cavity flow: Iso vorticity lines whose values are the
same as those in [Kohno and Bathe (2006)]
5 Concluding remarks
This paper is concerned with the use of local integrated RBFNs and
Cartesian grids in the point-collocation and control-volume
frameworks. Two main advantages of the present local techniques are
that (i) their matrices are sparse and (ii) their preprocessing is
simple. Numerical results show that (i) both local IRBFN meth- ods
result in the system matrix with a much lower condition number than
global RBF techniques, (ii) they outperform standard control-volume
techniques regard- ing accuracy for a given grid size, (iii) the
local IRBFN control-volume technique is much more accurate than the
local IRBFN collocation technique, (iv) the local IRBFN
control-volume technique has the capability to produce accurate
results for the simulation of flow problems having steep gradients
and complex patterns.
234 Copyright © 2009 Tech Science Press CMES, vol.51, no.3,
pp.213-238, 2009
−1.5 −1 −0.5 0 0.5 1 1.5 0
0.5
1
1.5
2
2.5
3
Re = 100
0.5
1
1.5
2
2.5
3
Re = 200
0.5
1
1.5
2
2.5
3
v 2 by FEMPSfrag repla ements
Re = 500 Figure 11: Cavity flow: Vertical and horizontal velocity
profiles along the centre line (x1 = 0) and the horizontal line (x2
= 2) for three Reynolds numbers. Results by FEM [Kohno and Bathe
(2006)] are also included for comparison purposes. It is noted that
the obtained results on the two grids used are indiscernible.
Cartesian-Grid Local-IRBFN Scheme 235
Acknowledgement: This work is supported by the Australian Research
Council. We would like to thank the referees for their helpful
comments.
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